Abstract

This paper deals with the asymptotic behavior of positive solutions of certain forced fractional differential equations of the form , , , where , , and is a real constant. From the obtained results, we derive a technique which can be applied to some related fractional differential equations.

1. Introduction

Consider the forced fractional differential equation where , , and is a real constant.

In the sequel we assume that(i) is a continuous function;(ii) is continuous and assume that there exists a continuous function;  and a real number, , and is a real number such that

We only consider those solutions of (1) that are continuable and nontrivial in any neighborhood of . Such solution is said to be oscillatory if there exists a sequence , such that , and it is nonoscillatory otherwise.

In the last few decades, integral, integrodifferential, and fractional differential equations have gained considerably more attention due to their applications in many engineering and scientific disciplines as the mathematical models for systems and processes in fields of physics, mechanics, chemistry, aerodynamics, and the electrodynamics of complex media. For more details one can refer to [1–8].

Oscillation and asymptotic behavior results for integral, integrodifferential, and fractional differential equations are scarce; some results can be found in [5, 9–13]. It seems that there are no such results for forced fractional differential equations of type (1). The main objective of this paper is to establish some new criteria on the asymptotic behavior of positive solutions of (1) which is equivalent to the Volterra type equation:From the obtained results, we derive a technique which can be applied to some related fractional differential equations.

We note that is the Caputo derivative of the order of a -scalar valued function defined on the interval , . For the case when , this definition has been given by Caputo [14]. For the definition of the Caputo derivative of order , , see [1, 15, 16].

2. Main Results

To obtain our main results of this paper, we need the following two lemmas.

Lemma 1 (see [5, 7]). Let , , and be positive constants such that , . Then where , , , and .

Lemma 2 (see [17]). If and are nonnegative, thenwhere equality hold if and only if .

In what follows, we let and , for some , where is a continuous function.

Now we give sufficient conditions under which any solution of (1) satisfies as .

Theorem 3. Let and suppose that , , and , , , and is bounded on ,where the function is defined as in (7) for any . If is a positive solution of (1), then .

Proof. Let be an eventually positive solution of (1). We may assume that for for some We let . In view of (i) and (ii) we may then write Applying (6) of Lemma 2 to withwe haveand hence we obtainIntegrating inequality (15) from to and interchanging the order of integration, one can easily obtainInterchanging the order of integration in second integral, we havewhere is the upper bound of the function .
Integrating (17) from to and interchanging the order of integration in the last integral, we findNow, one can easily see that where . From the hypotheses of the theorem, we see thatwith as the upper bound of the functionApplying Holder’s inequality and Lemma 1, we obtain where and and soThus, inequality (20) becomesUsing this inequality and the elementary inequality we obtain from (24) If we denote , that is, , , and , then The conclusions follow from Gromwell’s inequality and conclude thatThis completes the proof.

Remark 4. Condition (10) can be replaced byand the result remains valid.

The following example is illustrative.

Example 5. Let , , , and . Clearly Let the functions and be as in (i) and (ii) with being a bounded function and let , , and , where and is a continuous function with ,Condition (10) is also fulfilled. Thus, all conditions of Theorem 3 are satisfied and hence every positive solution of (1) satisfies .
Next, we consider the fractional differential equation where , , and is a real constant.
Now we give sufficient conditions under which any positive solution of (32) satisfies .

Theorem 6. Let and suppose that , , and , , and ,for any . If is a positive solution of (32), then .

Proof. Let be an eventually positive solution of (32). We may assume that for for some We let . In view of (ii) we may then writeProceeding as in the proof of Theorem 3, we obtain Integrating inequality (37) from to and interchanging the order of integration, one can easily obtainInterchanging the order of integration in second integral we haveThe rest of the proof is similar to that of Theorem 3 and hence is omitted.